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        Simulation: Vehicle
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                <h1 id="vehicle">Vehicle</h1>
<div class="admonition example">
<p class="admonition-title">Example</p>
<p><code class="codehilite"><span class="err">Vehicle/</span></code></p>
</div>
<h2 id="problem-description">Problem Description</h2>
<ul>
<li>
<p>The state vector of the vehicle is <span><span class="MathJax_Preview">x =  [y,z  , v ,\theta]^{T}\in\mathbb{R}^{4}</span><script type="math/tex">x =  [y,z  , v ,\theta]^{T}\in\mathbb{R}^{4}</script></span>, where <span><span class="MathJax_Preview">(y,z)</span><script type="math/tex">(y,z)</script></span> is the position coordinate, <span><span class="MathJax_Preview">v</span><script type="math/tex">v</script></span> is the velocity, and <span><span class="MathJax_Preview">\theta</span><script type="math/tex">\theta</script></span> is the heading angle. </p>
</li>
<li>
<p>The input vector is <span><span class="MathJax_Preview">u=[F,s]^T</span><script type="math/tex">u=[F,s]^T</script></span>, where <span><span class="MathJax_Preview">F</span><script type="math/tex">F</script></span> is the acceleration force and <span><span class="MathJax_Preview">s</span><script type="math/tex">s</script></span> is the steering torque. The control input is bounded by <span><span class="MathJax_Preview">[-5,-1]^T \leq u \leq [5,1]^T</span><script type="math/tex">[-5,-1]^T \leq u \leq [5,1]^T</script></span>.</p>
</li>
<li>
<p>The vehicle starts from position <span><span class="MathJax_Preview">(-2,0)</span><script type="math/tex">(-2,0)</script></span>. It's aim is to reach <span><span class="MathJax_Preview">(3.5,2)</span><script type="math/tex">(3.5,2)</script></span> while avoiding two obstacles with positions <span><span class="MathJax_Preview">(0,0)</span><script type="math/tex">(0,0)</script></span> and <span><span class="MathJax_Preview">(2,2)</span><script type="math/tex">(2,2)</script></span>. These constraints are formulated as <span><span class="MathJax_Preview">y^2+z^2\geq 1</span><script type="math/tex">y^2+z^2\geq 1</script></span> and <span><span class="MathJax_Preview">(y-2)^2+(z-2)^2\geq 1</span><script type="math/tex">(y-2)^2+(z-2)^2\geq 1</script></span>.</p>
</li>
<li>
<p>The dynamics of the vehicle are given by the following equations (<span><span class="MathJax_Preview">m</span><script type="math/tex">m</script></span>: mass and <span><span class="MathJax_Preview">I</span><script type="math/tex">I</script></span>: moment of inertia):</p>
</li>
</ul>
<div>
<div class="MathJax_Preview">\begin{align}
&amp;\dot{y} = v\cos(\theta)\\
&amp;\dot{z} = v\sin(\theta)\\
&amp;\dot{v} = F/m\\
&amp;\dot{\theta}  =  s/I
\end{align}</div>
<script type="math/tex; mode=display">\begin{align}
&\dot{y} = v\cos(\theta)\\
&\dot{z} = v\sin(\theta)\\
&\dot{v} = F/m\\
&\dot{\theta}  =  s/I
\end{align}</script>
</div>
<h2 id="ocp-in-parnmpc">OCP in <strong>ParNMPC</strong></h2>
<ul>
<li>
<p>The nonlinear constraints are transfered into <span><span class="MathJax_Preview">y^2+z^2 =  C_{1}</span><script type="math/tex">y^2+z^2 =  C_{1}</script></span> and <span><span class="MathJax_Preview">(y-2)^2+(z-2)^2 =  C_{2}</span><script type="math/tex">(y-2)^2+(z-2)^2 =  C_{2}</script></span> with <span><span class="MathJax_Preview">C_{1}\geq 1</span><script type="math/tex">C_{1}\geq 1</script></span> and <span><span class="MathJax_Preview">C_{2}\geq 1</span><script type="math/tex">C_{2}\geq 1</script></span>.</p>
</li>
<li>
<p>The state constraints  <span><span class="MathJax_Preview">C_{1}\geq 1</span><script type="math/tex">C_{1}\geq 1</script></span> and <span><span class="MathJax_Preview">C_{2}\geq 1</span><script type="math/tex">C_{2}\geq 1</script></span> are softened by introducing a positive slack variable.</p>
</li>
</ul>
<h2 id="closed-loop-simulation-using-parnmpc">Closed-loop Simulation using <strong>ParNMPC</strong></h2>
<p>Closed-loop simulation result:
<p align="center">
<img src="../vehicle.png" width="400px">
</p></p>
<h3 id="step-1-nmpc-problem-formulation">Step 1. NMPC problem formulation</h3>
<p>See <a href="../workflow/#nmpc_formulation">Workflow of ParNMPC &gt; NMPC Problem Formulation</a>.</p>
<div class="admonition example">
<p class="admonition-title">Example</p>
<p><code class="codehilite"><span class="err">Vehicle/NMPC_Problem_Formulation.m</span></code></p>
</div>
<h3 id="step-2-code-generation-and-deployment-in-simulink">Step 2. Code generation and deployment in Simulink</h3>
<p>See <a href="../workflow/#workflow_deploy_simulink">Workflow of ParNMPC &gt; Code Generation and Deployment &gt; Simulink</a>.</p>
<ol>
<li>
<p>Code generation</p>
<div class="admonition example">
<p class="admonition-title">Example</p>
<p><code class="codehilite"><span class="err">Vehicle/Simu_Simulink_Setup.m</span></code></p>
</div>
</li>
<li>
<p>Deployment</p>
<div class="admonition example">
<p class="admonition-title">Example</p>
<p><code class="codehilite"><span class="err">Vehicle/Simu_Simulink.slx</span></code></p>
</div>
</li>
</ol>
                
                  
                
                
              
              
                


              
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